<style>
    .btn-control {
        font: bold 12px/20px 'Helvetica Neue', Arial, Helvetica, sans-serif;
        background-color: #3386c0;
        color: #fff;
        position: absolute;
        top: 20px;
        left: 50%;
        z-index: 1;
        border: none;
        width: 200px;
        margin-left: -100px;
        display: block;
        cursor: pointer;
        padding: 10px 20px;
        border-radius: 3px;
    }

    .btn-control:hover {
        background-color: #4ea0da;
    }
</style>
<div id="map"></div>
<button id="zoomto" class="btn-control">Zoom to bounds</button>

<script>
    // A GeoJSON object with a LineString route from the White House to Capitol Hill
    var geojson = {
        'type': 'FeatureCollection',
        'features': [
            {
                'type': 'Feature',
                'geometry': {
                    'type': 'LineString',
                    'properties': {},
                    'coordinates': [
                        [-77.0366048812866, 38.89873175227713],
                        [-77.03364372253417, 38.89876515143842],
                        [-77.03364372253417, 38.89549195896866],
                        [-77.02982425689697, 38.89549195896866],
                        [-77.02400922775269, 38.89387200688839],
                        [-77.01519012451172, 38.891416957534204],
                        [-77.01521158218382, 38.892068305429156],
                        [-77.00813055038452, 38.892051604275686],
                        [-77.00832366943358, 38.89143365883688],
                        [-77.00818419456482, 38.89082405874451],
                        [-77.00815200805664, 38.88989712255097]
                    ]
                }
            }
        ]
    };

    var map = new mapboxgl.Map({
        container: 'map',
        style: 'mapbox://styles/mapbox/light-v10',
        center: [-77.0214, 38.897],
        zoom: 12
    });

    map.on('load', function() {
        map.addSource('LineString', {
            'type': 'geojson',
            'data': geojson
        });
        map.addLayer({
            'id': 'LineString',
            'type': 'line',
            'source': 'LineString',
            'layout': {
                'line-join': 'round',
                'line-cap': 'round'
            },
            'paint': {
                'line-color': '#BF93E4',
                'line-width': 5
            }
        });

        document.getElementById('zoomto').addEventListener('click', function() {
            // Geographic coordinates of the LineString
            var coordinates = geojson.features[0].geometry.coordinates;

            // Pass the first coordinates in the LineString to `lngLatBounds` &
            // wrap each coordinate pair in `extend` to include them in the bounds
            // result. A variation of this technique could be applied to zooming
            // to the bounds of multiple Points or Polygon geomteries - it just
            // requires wrapping all the coordinates with the extend method.
            var bounds = coordinates.reduce(function(bounds, coord) {
                return bounds.extend(coord);
            }, new mapboxgl.LngLatBounds(coordinates[0], coordinates[0]));

            map.fitBounds(bounds, {
                padding: 20
            });
        });
    });
</script>
